sp.spectral theory - If the first Dirichlet eigenfunction on a set $D$ is regular up to the boundary, is $D$ regular?
Given any open set $D$ in $\mathbb R^n$, we can define the first Dirichlet eigenfunction $u$ of $-\Delta$ on $D$ as the minimizer of the Rayleigh quotient over $H_0^1(D)$. Interior regularity of $u$ follows easily, but in general it seems we cannot say anything about the regularity of $u$ up to the boundary.
But conversely, if we know $u$ is say globally Lipschitz continuous on the closure of $D$ (assume $D$ is bounded), does this tell us anything about the regularity of the boundary $\partial D$? For example, that it is locally a Lipschitz graph? What if $u$ has even more regularity?
This seems like a natural question, but I cannot find a reference or think of a way to construct counterexamples.